#7711227

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	#38425	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4762 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618423);