#6370451

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,h^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf,e^d=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277647);