#396913

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1076 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)(7,9)(8,10)(11,12)(13,14,19,22)(15,16,20,18)(17,25,27,21)(23,26,28,24);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,18,14,19,20,16,22)(17,23,26,21,27,28,24,25);(2,4)(6,10)(9,12)(13,16,19,18)(14,20,22,15)(17,24,27,26)(21,28,25,23);(13,17,18,26,19,27,16,24)(14,21,20,28,22,25,15,23)

Polycyclic presentation:

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

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, 304109);