#7628648

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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	#35261	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5246 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,27)(28,32)(29,31);(2,6)(7,8)(17,19)(18,23)(20,29)(21,28)(22,27)(24,32)(25,26)(30,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(19,27)(23,26)(28,29)(31,32);(1,4,5,3)(2,8,6,7)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,31,26,32);(17,21,22,20)(18,24,25,30)(19,28,27,29)(23,32,26,31);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535844);