#7622564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6370843
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#117
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂²×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,10,5,11,16,25,6,7,15,19,17,8,20,22)(3,9,13,23,14,24,28,32,12,18,26,29,27,21,30,31);(2,7)(4,15)(8,11)(9,18)(13,26)(16,20)(21,24)(28,30);(2,8,7,11)(4,15)(9,21,18,24)(10,25,19,22)(13,26)(16,20)(23,32,29,31)(28,30);(1,3)(2,9)(4,13)(5,14)(6,12)(7,18)(8,21)(10,23)(11,24)(15,26)(16,28)(17,27)(19,29)(20,30)(22,31)(25,32);(3,12)(9,18)(13,26)(14,27)(21,24)(23,29)(28,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215833, #72219678, #72219679
maximal normal subgroup of	#72215833, #72219678, #72219679
maximal quotient of	#72226782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529760);