#495080

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1948
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285601, #72285921, #72285937
maximal normal subgroup of	#72285601, #72285921, #72285937
maximal quotient of	#72310101, #72321788, #72342539

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402276);