#247703

order	512
nilpotency class	3
isoclinism	isoclinic class of #247687
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

1587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|b^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=gh,c^b=cg,e^b=eh,e^c=eh,f^c=fi,e^d=eh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214667, #72287110
maximal normal subgroup of	#72214667, #72287110
maximal quotient of	#72278794, #72278842

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154899);