#7629116

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
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	#34977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5189 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)(9,10,13,15)(11,14,12,16)(17,18,22,21)(19,23,27,26)(20,24,29,30)(25,31,28,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,26)(22,27)(24,32)(25,29)(30,31);(10,15)(14,16)(17,20,22,29)(18,24,21,30)(19,25,27,28)(23,31,26,32);(17,21,22,18)(19,26,27,23)(20,24,29,30)(25,31,28,32);(9,12,13,11)(10,16,15,14)(19,27)(23,26)(25,28)(31,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=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, 7536312);