#7500942

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3453 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,g^c=gi,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315519, #72315621
maximal normal subgroup of	#72315519, #72315621

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408138);