#6356592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356533
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=i,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,g^c=gi,e^d=ehi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339745, #72339754, #72339773, …
maximal normal subgroup of	#72339745, #72339754, #72339773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263788);