#7679671

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
prime radical	#38041	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4653 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7586867);