#3402814

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1775 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3310010);