#1681296

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1710 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=fgh,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1588492);