#7638648

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4624 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi

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, 7545844);