#117038

order	512
nilpotency class	3
isoclinism	isoclinic class of #117035
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24234);