#7062401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	#39722	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1929 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6969597);