#216173

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=ehi,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,g^c=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123369);