#389404

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

922 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)(7,11)(8,10)(9,12)(13,14)(15,25,28,41)(16,29)(17,32,34,43)(18,23)(19,24)(20,27,39,35)(21,31)(22,33,37,36)(26,38)(30,40)(42,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,19,28)(14,20,24,39)(16,17,31,34)(18,35,30,27)(21,37,29,22)(23,41,40,25)(26,36,44,33)(32,38,43,42);(2,4)(6,10)(9,11)(13,16,30,44,19,31,18,26)(14,21,40,38,24,29,23,42)(15,17,27,33,28,34,35,36)(20,37,25,43,39,22,41,32);(7,12)(9,11)(13,17)(14,22)(15,26)(16,27)(18,36)(19,34)(20,38)(21,41)(23,43)(24,37)(25,29)(28,44)(30,33)(31,35)(32,40)(39,42)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=efh,e^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,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, 296600);