#6364100

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
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

1622 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)(3,9)(4,13)(5,7)(6,12)(8,14)(10,16)(11,15)(17,18,24,27)(19,30,25,32)(20,31,21,26)(22,29,23,28);(1,3,5,10)(2,6,7,15)(4,11,14,12)(8,16,13,9)(17,19,24,25)(18,20,27,21)(22,31,23,26)(28,32,29,30);(3,10)(6,15)(9,16)(11,12)(17,20,24,21)(18,25,27,19)(22,30,23,32)(26,28,31,29);(3,11)(4,14)(6,16)(8,13)(9,15)(10,12)(17,21,24,20)(18,26,27,31)(19,29,25,28)(22,30,23,32);(2,7)(6,15)(8,13)(9,16)(18,27)(19,25)(26,31)(28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306857, #72306861, #72306890
maximal normal subgroup of	#72306857, #72306861, #72306890

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271296);