#354822

order	512
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#379
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	C₄×#379
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3465 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)(4,19,22,10)(5,11)(6,27)(7,12)(8,13)(9,17)(15,39,42,37)(16,38)(18,32)(20,46,49,33)(21,34,23,47)(24,52)(25,35)(26,36)(28,31)(29,41,45,51)(30,43)(40,59,61,58)(44,56)(48,57,50,63)(53,55)(54,60,62,64);(1,3,7,17,8,18,6,28)(2,9,12,32,13,31,27,14)(4,15,23,41,22,42,21,51)(5,16,25,43,26,44,24,53)(10,29,34,39,19,45,47,37)(11,30,35,56,36,55,52,38)(20,40,50,60,49,61,48,64)(33,54,57,59,46,62,63,58);(1,4)(2,10)(3,15)(5,20)(6,21)(7,23)(8,22)(9,29)(11,33)(12,34)(13,19)(14,37)(16,40)(17,41)(18,42)(24,48)(25,50)(26,49)(27,47)(28,51)(30,54)(31,45)(32,39)(35,57)(36,46)(38,58)(43,60)(44,61)(52,63)(53,64)(55,62)(56,59);(1,5,7,25,8,26,6,24)(2,11,12,35,13,36,27,52)(3,16,17,43,18,44,28,53)(4,20,23,50,22,49,21,48)(9,30,32,56,31,55,14,38)(10,33,34,57,19,46,47,63)(15,40,41,60,42,61,51,64)(29,54,39,59,45,62,37,58)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285688, #72286021, #72317336, …
maximal normal subgroup of	#72285688, #72286021, #72317336, …
maximal quotient of	C₂×#354822, #72342992

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262018);