#363177

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2408 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)(3,4)(13,14,18,22,17,21,30,37)(15,19,25,34,24,33,40,43)(16,26,29,38,28,20,31,36)(23,35,39,44,27,32,41,42);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,24)(18,25)(20,35)(21,33)(22,34)(23,28)(26,32)(29,41)(30,40)(31,39)(36,44)(37,43)(38,42);(6,9)(10,12)(13,16,17,28)(14,20,21,26)(15,23,24,27)(18,31,30,29)(19,32,33,35)(22,38,37,36)(25,41,40,39)(34,44,43,42);(2,4)(5,7,8,11)(6,10,9,12)(15,24)(19,33)(23,27)(25,40)(32,35)(34,43)(39,41)(42,44)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=gi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,h^c=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, 270373);