#609198

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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,e^2=g,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	#72239620, #72279674, #72281522, …
maximal normal subgroup of	#72239620, #72279674, #72281522, …
maximal quotient of	#72239728

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516394);