#6369108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369070
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

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306666, #72306692, #72306716
maximal normal subgroup of	#72306666, #72306692, #72306716

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276304);