#353608

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13753
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14,19,39)(4,20)(5,11)(6,12)(7,27)(8,13)(9,17,31,18)(10,22)(15,29,42,54)(16,38,45,59)(21,48)(23,33)(24,35)(25,51)(26,36)(28,34)(30,43,55,44)(32,49)(37,46,47,41)(40,53,61,64)(50,56)(52,57)(58,62,63,60);(1,3)(2,9,13,31)(4,15)(5,16)(6,17)(7,18)(8,19)(10,29,34,54)(11,30,36,55)(12,14,27,39)(20,47,33,37)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(28,46)(32,53,57,64)(35,38,51,59)(48,63,56,58)(49,60)(50,61)(52,62);(1,4)(2,10)(3,15)(5,21)(6,22)(7,28)(8,23)(9,29)(11,32)(12,33)(13,34)(14,37)(16,40)(17,41)(18,46)(19,42)(20,27)(24,49)(25,52)(26,50)(30,53)(31,54)(35,56)(36,57)(38,58)(39,47)(43,60)(44,62)(45,61)(48,51)(55,64)(59,63);(1,5,6,24,8,26,7,25)(2,11,12,35,13,36,27,51)(3,16,17,43,19,45,18,44)(4,21,22,49,23,50,28,52)(9,30,14,38,31,55,39,59)(10,32,33,56,34,57,20,48)(15,40,41,60,42,61,46,62)(29,53,37,58,54,64,47,63)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,g^b=gi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285493, C₂×#353608, #72285922, …
maximal normal subgroup of	#72285493, C₂×#353608, #72285922, …
maximal quotient of	C₂×#353608

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260804);