#6811380

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42652	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2059 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)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14,17,21)(15,23,26,19)(16,28,30,20)(18,22,25,32)(24,35,37,31)(27,33,40,36)(29,41,39,34)(38,42,44,43);(5,7,10,6)(8,12,9,11)(15,24)(16,29)(18,25)(19,31)(20,34)(22,32)(23,35)(26,37)(27,40)(28,41)(30,39)(33,36);(2,4)(13,15,25,37,17,26,18,24)(14,19,32,35,21,23,22,31)(16,27,39,44,30,40,29,38)(20,33,41,43,28,36,34,42);(5,8,10,9)(6,11,7,12)(15,26)(19,23)(24,37)(27,40)(31,35)(33,36)(38,44)(42,43);(5,9,10,8)(6,11,7,12)(13,16,18,29,17,30,25,39)(14,20,22,34,21,28,32,41)(15,27,24,38,26,40,37,44)(19,33,31,42,23,36,35,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718576);