#775399

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3816 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)(9,10,13,16)(11,14,12,15)(17,18,21,20)(19,22,24,23);(6,8)(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,21)(22,24);(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24);(1,2)(3,4)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24);(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,23,24,22)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297345, #72297409, #72297424, …
maximal normal subgroup of	#72297345, #72297409, #72297424, …
maximal quotient of	#72298267, #72332965

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 682595);