#390397

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 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)(7,11)(8,10)(9,12)(13,14,18,22,19,23,17,31)(15,24,26,20,27,35,32,36)(16,28,30,21,29,38,33,37)(25,34,42,44,40,43,41,39);(2,4)(5,7)(6,9)(8,12)(10,11)(16,29)(21,37)(25,40)(28,38)(30,33)(34,43)(39,44)(41,42);(1,2,3,4)(13,15,18,26,19,27,17,32)(14,20,22,35,23,36,31,24)(16,25,30,42,29,40,33,41)(21,34,38,44,37,43,28,39);(6,10)(7,12)(13,16)(14,21)(15,25)(17,33)(18,30)(19,29)(20,34)(22,38)(23,37)(24,39)(26,42)(27,40)(28,31)(32,41)(35,44)(36,43)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,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, 297593);