#7680620

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

4631 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)(5,8)(6,7)(10,13)(14,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(1,3,6,5)(2,4,7,8)(9,10,12,13)(11,14,15,16)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30);(2,7)(4,8)(18,19)(20,27)(24,26)(28,32);(1,4,6,8)(2,3,7,5)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,3,6,5)(2,4,7,8)(9,11,12,15)(10,14,13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(1,5,6,3)(2,8,7,4)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 7587816);