#378987

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=ef,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,g^c=gi

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, 286183);