#7503446

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410642);