#366804

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=e,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274000);