#398673

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1112 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,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, 305869);