#7500984

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3452 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)(11,19)(12,22)(13,17)(14,18)(15,23)(16,24)(20,21);(1,2)(3,4)(5,6)(7,8)(9,10,14,18)(11,20,16,12)(13,23,15,17)(19,22,24,21);(6,8)(9,11,14,16)(10,12,18,20)(13,22,15,21)(17,24,23,19);(2,4)(9,12)(10,16)(11,18)(13,19)(14,20)(15,24)(17,22)(21,23);(9,13,14,15)(10,17,18,23)(11,21,16,22)(12,19,20,24)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315498, #72315519, #72315564
maximal normal subgroup of	#72315498, #72315519, #72315564

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408180);