#190985

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1808 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)(3,6)(4,5)(7,8)(9,10)(11,15,16,14)(12,13)(17,18)(19,21)(20,23)(22,24);(2,5)(6,8)(10,13)(14,15);(9,11)(10,14)(12,16)(13,15)(18,21)(19,24)(22,23);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=g,d^2=ehi,f^2=i,b^a=be,c^a=cf,f^a=fh,f^c=fi,g^c=ghi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309208, #72309237, #72309302
maximal normal subgroup of	#72309208, #72309237, #72309302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98181);