#7502134

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2816 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)(9,10,13,17)(11,18,20,22)(12,14,21,16)(15,23,24,19);(5,6)(7,8)(10,14)(11,19)(12,21)(16,17)(18,22)(20,23);(2,4)(6,8);(2,4)(9,11,12,19,13,20,21,23)(10,15,16,18,17,24,14,22);(11,20)(15,24)(18,22)(19,23)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312904, #72312915, #72313009
maximal normal subgroup of	#72312904, #72312915, #72313009

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409330);