#8108431

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

6130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261815, #72261907, #72261948
maximal normal subgroup of	#72261815, #72261907, #72261948

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015627);