#7674823

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	C₂×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4972 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,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, 7582019);