#7701816

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,15,13,22)(12,19,21,16)(18,24,20,23);(2,5)(6,8)(9,11,14,13)(10,15,17,22)(12,18,21,20)(16,23,19,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,24,19,23)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(9,13,14,11)(10,15,17,22)(12,20,21,18)(16,23,19,24)(27,35)(30,33)(34,40)(38,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=hi,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, 7609012);