#247952

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	#36128	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1410 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,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	#72338828, #72338833, #72338838
maximal normal subgroup of	#72338828, #72338833, #72338838

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155148);