#120681

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

498 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)(13,14,18,24,19,22,17,23)(15,25,27,37,28,20,32,35)(16,29,30,36,31,38,33,21)(26,40,42,39,43,44,41,34);(1,2)(3,4)(5,6,9,11,10,12,8,7)(13,15,17,27,19,28,18,32)(14,20,23,35,22,25,24,37)(16,26,30,41,31,43,33,42)(21,34,29,44,36,39,38,40);(2,4)(5,7,9,6,10,11,8,12)(13,16,19,31)(14,21)(15,26,28,43)(17,30,18,33)(20,34)(22,36)(23,29)(24,38)(25,39)(27,41,32,42)(35,44)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,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, 27877);