#845641

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	#15320	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:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=f,e^2=fg,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, 752837);