#395003

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
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	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1660 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)(13,14,19,17)(15,20,18,23)(16,24,26,28)(21,27,25,22);(1,2)(3,4)(6,9)(10,12)(13,15,19,18)(14,20,17,23)(16,22,26,27)(21,28,25,24);(2,4)(13,16)(14,21)(15,22)(17,25)(18,27)(19,26)(20,28)(23,24);(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,23,18,20)(16,25,26,21)(22,24,27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^c=fi,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, 302199);