#6366333

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366206
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1569 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340115, #72340124, #72340135
maximal normal subgroup of	#72340115, #72340124, #72340135

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273529);