#7701452

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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	#46380	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4855 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)(11,15)(12,16)(13,14)(17,18,22,25)(19,26,28,23)(20,24,21,30)(27,32,29,31);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14);(10,14)(11,12)(17,19,22,28)(18,23,25,26)(20,29,21,27)(24,32,30,31);(9,11,13,12)(10,15,14,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(9,12,13,11)(10,15,14,16)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7608648);