#248263

order	512
nilpotency class	3
isoclinism	isoclinic class of #248263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 16 points:

(1,2)(3,5)(4,7)(6,8)(10,14)(11,12)(15,16);(3,8)(5,6)(9,10)(11,16)(12,14)(13,15);(2,5)(3,8)(6,7)(9,10)(11,14)(12,16)(13,15);(9,10)(11,16)(12,14)(13,15)

Polycyclic presentation:

2^9|e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fhi,f^c=fi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214707, #72258112, #72258489
maximal normal subgroup of	#72214707, #72258112
maximal quotient of	#72228199

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155459);