#7647094

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	#38908	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4663 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7554290);