#120682

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(6,7)(8,9)(11,12)(13,14,17,22,18,21,24,33)(15,23,27,35,28,19,26,36)(16,29,31,20,32,38,30,37)(25,40,41,34,42,44,43,39);(1,2)(3,4)(5,6,8,7,10,12,9,11)(13,15,24,27,18,28,17,26)(14,19,33,36,21,23,22,35)(16,25,31,43,32,42,30,41)(20,34,38,40,37,39,29,44);(2,4)(5,7,9,6,10,11,8,12)(13,16,18,32)(14,20)(15,25,28,42)(17,30,24,31)(19,34)(21,37)(22,29)(23,39)(26,41,27,43)(33,38)(35,44)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27878);