#7674757

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4640 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=gh,e^2=g,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, 7581953);