#393378

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,g^b=gi,h^b=hi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300574);