#7702862

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	#45451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4899 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)(12,14)(13,16)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(2,5)(6,8)(17,19,22,21)(18,23,25,29)(20,27,28,26)(24,31,32,30);(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,32)(29,31);(11,16)(13,15)(17,21,22,19)(18,23,25,29)(20,26,28,27)(24,31,32,30);(10,14)(13,15)(20,28)(24,32)(26,27)(30,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=i,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,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, 7610058);