#879253

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

3138 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)(10,12)(11,15)(18,22)(20,23);(1,2,3,4)(6,8)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,24)(21,22);(1,2)(3,4)(5,6)(7,8);(1,2)(3,4)(9,11)(10,14)(12,16)(13,15)(17,19,21,24)(18,23,22,20);(9,12,13,10)(11,16,15,14)(17,20,21,23)(18,24,22,19)

Polycyclic presentation:

2^9|a^2=g,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, 786449);