#7669974

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4829 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)(3,6)(4,5)(7,8)(9,10,13,11)(12,15,16,14)(17,18,21,25)(19,22,26,23)(20,28,30,24)(27,32,29,31);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(17,19,21,26)(18,22,25,23)(20,27,30,29)(24,31,28,32);(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,29)(21,26)(22,25)(24,31)(27,30)(28,32);(12,16)(14,15)(19,26)(22,23)(27,29)(31,32);(17,20,21,30)(18,24,25,28)(19,27,26,29)(22,31,23,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577170);