#215616

order	512
nilpotency class	3
isoclinism	isoclinic class of #184792
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1546 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=eh,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122812);