#6376274

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6283470);