#358758

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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)(7,11)(8,10)(9,12)(13,14,17,23,19,24,18,22)(15,25,28,20,27,36,29,35)(16,30,31,38,33,37,32,21)(26,34,41,43,40,44,42,39);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,28,19,27,18,29)(14,20,22,25,24,35,23,36)(16,26,31,41,33,40,32,42)(21,34,37,39,38,44,30,43);(2,4)(5,7,9,8)(6,11,12,10)(13,16,17,31,19,33,18,32)(14,21,22,37,24,38,23,30)(15,26,28,41,27,40,29,42)(20,34,25,39,35,44,36,43);(5,8,9,7)(6,10,12,11)(15,27)(20,35)(25,36)(26,40)(28,29)(34,44)(39,43)(41,42)

Polycyclic presentation:

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

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, 265954);