#414962

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1446 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)(7,11)(8,9)(10,12)(13,14)(15,22,25,27)(16,21)(17,20)(18,23,19,24)(26,28);(1,2)(3,4)(13,15)(14,18)(16,24)(17,25)(19,20)(21,27)(22,28)(23,26);(2,4)(6,9)(10,11)(13,15)(14,19)(16,23)(17,25)(18,20)(21,27)(22,28)(24,26);(5,7,8,12)(6,10,9,11)(14,20)(18,19)(21,28)(22,27)

Polycyclic presentation:

2^9|a^2=efh,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^c=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, 322158);