#7692389

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4592 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7599585);