#396033

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^c=hi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303229);