#247906

order	512
nilpotency class	3
isoclinism	isoclinic class of #247879
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
prime radical	#61559	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282337, #72282373
maximal normal subgroup of	#72282337, #72282373
maximal quotient of	#72282948, #72283003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155102);