#371304

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1261 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,d^2=gi,e^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,e^d=ei,f^d=fi,h^d=hi

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, 278500);