#363280

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2430 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)(13,14,19,24,20,25,18,36)(15,26,28,21,17,34,35,23)(16,29,32,40,33,22,31,39)(27,41,42,38,30,44,43,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,27)(17,20)(18,35)(19,28)(22,37)(23,25)(24,34)(26,36)(29,38)(30,33)(31,43)(32,42)(39,41)(40,44);(6,9)(10,12)(13,16,20,33)(14,22,25,29)(15,27,17,30)(18,32,19,31)(21,37,23,38)(24,40,36,39)(26,41,34,44)(28,43,35,42);(2,4)(5,7,8,11)(6,10,9,12)(13,17,20,15)(14,23,25,21)(16,30,33,27)(18,28,19,35)(22,38,29,37)(24,26,36,34)(31,42,32,43)(39,44,40,41)

Polycyclic presentation:

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

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, 270476);