#210282

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#18522	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117478);