#7744212

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	#37559	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4572 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7651408);