#7669294

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
prime radical	#34741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4719 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)(9,10,13,12)(11,15,16,14)(17,18,22,25)(19,23,28,32)(20,24,30,21)(26,31,29,27);(2,5)(6,8)(9,10,13,12)(11,14,16,15)(17,19,22,28)(18,23,25,32)(20,29,30,26)(21,31,24,27);(9,11,13,16)(10,14,12,15)(17,20)(18,24)(19,26)(21,25)(22,30)(23,31)(27,32)(28,29);(11,16)(14,15)(17,21,22,24)(18,20,25,30)(19,27,28,31)(23,26,32,29);(9,12,13,10)(11,15,16,14)(18,25)(20,30)(23,32)(26,29);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=i,e^2=h,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, 7576490);