#364706

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1560 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)(7,10)(8,9)(11,12)(13,14,18,24,19,23,17,22)(15,20,27,37,28,36,26,35)(16,29,31,40,33,21,32,39)(25,41,42,38,43,34,30,44);(1,2)(3,4)(6,9)(10,12)(13,15,18,27,19,28,17,26)(14,20,24,37,23,36,22,35)(16,30,31,25,33,42,32,43)(21,38,39,34,29,44,40,41);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,31)(18,32)(19,33)(20,34)(22,39)(23,29)(24,40)(26,42)(27,30)(28,43)(35,44)(36,41)(37,38);(6,9)(10,12)(13,17,19,18)(14,22,23,24)(15,26,28,27)(16,31,33,32)(20,35,36,37)(21,39,29,40)(25,42,43,30)(34,44,41,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271902);