#789380

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

3881 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)(6,8)(9,10,14,12)(11,16,13,15)(17,18,22,21)(19,24,23,20);(1,2,3,4)(9,11)(10,15)(12,16)(13,14);(9,10,14,12)(11,16,13,15)(17,19)(18,20)(21,24)(22,23);(1,2,3,4)(5,6)(7,8)(9,12,14,10)(11,16,13,15)(17,20,22,24)(18,23,21,19);(1,2)(3,4)(9,13,14,11)(10,15,12,16)(17,21,22,18)(19,24,23,20)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=h,d^2=fg,e^2=gh,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	#72279963, #72280075, #72280272, …
maximal normal subgroup of	#72279963, #72280075, #72280272, …
maximal quotient of	C₂×#789380

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 696576);