#120246

order	512
nilpotency class	3
isoclinism	isoclinic class of #98904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 27442);