#7712818

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

6384 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:

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

Polycyclic presentation:

2^9|d^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262195, #72262230
maximal normal subgroup of	#72262195, #72262230

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7620014);