#775185

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

3674 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)(9,10,14,13)(11,16,12,15)(17,18,21,23)(19,24,20,22);(6,8)(9,11,14,12)(10,15,13,16)(17,19)(18,22)(20,21)(23,24);(5,6,7,8)(9,11,14,12)(10,16,13,15)(17,18)(19,24)(20,22)(21,23);(1,2)(3,4)(9,12,14,11)(10,16,13,15)(17,18)(19,22)(20,24)(21,23);(9,13,14,10)(11,15,12,16)(17,20,21,19)(18,22,23,24)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fh,d^2=fg,e^2=f,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	#72297508, #72297547, #72297562, …
maximal normal subgroup of	#72297508, #72297547, #72297562, …
maximal quotient of	#72298306

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 682381);