#398606

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1191 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305802);