#7502322

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23391	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6,8,9)(7,11,12,10)(14,18)(15,19)(16,24)(20,25)(21,23)(22,28);(1,2)(3,4)(6,9)(7,12)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,26,28,27);(2,4)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(5,7,8,12)(6,10,9,11)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27);(15,23)(19,21)(22,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409518);