#6368786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368766
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

1548 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:

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

Polycyclic presentation:

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

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306692, #72306699, #72306700
maximal normal subgroup of	#72306692, #72306699, #72306700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275982);