#403360

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

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

2764 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)(13,14,17,20)(15,18,22,25)(16,19,23,26)(21,27,28,24);(5,6,8,10)(7,11,12,9)(13,15)(14,18)(16,21)(17,22)(19,24)(20,25)(23,28)(26,27);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,28)(25,27);(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,28)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268294
maximal normal subgroup of	#72268294

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310556);