#1747066

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

1988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=h,d^2=fi,e^2=fh,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, 1654262);