#879254

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15123	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3192 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 786450);