#608885

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(5,6)(7,8)(10,13)(14,16)(17,18)(19,22)(20,24)(21,23);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,22,23,24);(2,4)(17,20,21,19)(18,22,23,24);(6,8)(10,13)(14,16);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303648, #72303657, #72303672
maximal normal subgroup of	#72303648, #72303657, #72303672

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516081);