#693111

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2878 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fg,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 600307);