#459297

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2006	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1269 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)(7,11)(8,9)(10,12)(13,14,18,15)(16,24,17,20)(19,22,28,23)(21,27,25,26);(1,2,4,3)(6,9)(10,11)(13,15,18,14)(16,21,17,25)(19,22,28,23)(20,26,24,27);(2,3)(5,7)(6,10)(8,12)(9,11)(13,16,18,17)(14,20,15,24)(19,27,28,26)(21,23,25,22);(1,3,4,2)(7,12)(10,11)(13,17,18,16)(14,21,15,25)(19,27,28,26)(20,23,24,22)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366493);