#119585

order	512
nilpotency class	3
isoclinism	isoclinic class of #99180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=di,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^b=gh,e^c=ei,g^c=gh

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, 26781);