#8819019

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261733, #72261846, #72261985, …
maximal normal subgroup of	#72261733, #72261846, #72261985, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8726215);