#508276

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

793 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=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, 415472);