#7667765

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	C₂×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4971 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^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, 7574961);