#2762630

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

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, 2669826);