#686292

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4476 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)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24);(6,8)(9,11,13,16)(10,14,15,12)(18,21)(22,24);(5,6)(7,8)(10,15)(12,14)(17,19,20,23)(18,22,21,24);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24);(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)

Polycyclic presentation:

2^9|b^2=g,c^2=fg,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.

Relationships

maximal subgroup of	#72328097, #72328126, #72328186, …
maximal normal subgroup of	#72328097, #72328126, #72328186, …
maximal quotient of	#72329919, #72332609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 593488);