#120202

order	512
nilpotency class	3
isoclinism	isoclinic class of #120201
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,7,8,5)(4,6)(9,10,14,18,15,20,13,19)(11,21,24,33,25,32,23,16)(12,26,28,36,30,35,29,17)(22,31,27,34,38,40,39,37);(9,11,13,23,15,25,14,24)(10,16,18,21,20,33,19,32)(12,27,28,38,30,39,29,22)(17,34,35,31,36,37,26,40);(1,3)(2,5)(4,8)(6,7)(9,12,13,28,15,30,14,29)(10,17,18,35,20,36,19,26)(11,22,23,27,25,38,24,39)(16,31,21,37,33,40,32,34)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fh,f^c=fh,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, 27398);