#7708672

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	#37860	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=gh,e^2=gh,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, 7615868);