#8197015

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318322
maximal normal subgroup of	#72318322

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104211);