#232148

order	512
nilpotency class	3
isoclinism	isoclinic class of #232147
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#496	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

566 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,e^a=eh,c^b=cefi,d^b=dei,f^b=fh,g^b=gh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139344);