#7680645

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4643 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)(3,4)(5,7)(6,8)(10,13)(14,16)(17,18,22,23)(19,21,26,24)(20,25,30,31)(27,29,28,32);(1,3,6,5)(2,4,8,7)(9,10,12,13)(11,14,15,16)(18,23)(19,26)(25,31)(27,28);(1,3,6,5)(2,7,8,4)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,29,31,32);(1,4,6,7)(2,3,8,5)(17,20,22,30)(18,25,23,31)(19,27,26,28)(21,29,24,32);(1,3,6,5)(2,4,8,7)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,27)(22,26)(23,24)(25,29)(28,30)(31,32);(1,5,6,3)(2,7,8,4)(17,21)(18,19)(20,29)(22,24)(23,26)(25,27)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=hi,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, 7587841);