#380042

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1710 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=ei,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,d^c=de

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, 287238);