#2088652

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

2142 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)(5,6)(7,8)(9,11)(10,12)(14,17)(15,19)(21,22,25,29)(23,27,31,26)(24,28,33,36)(30,35,32,34);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,31)(22,26,29,27)(24,30,33,32)(28,34,36,35);(6,11)(7,12)(13,15,16,19)(14,18,17,20)(21,23)(22,27)(24,32)(25,31)(26,29)(28,34)(30,33)(35,36);(1,2)(3,4)(5,8,9,10)(6,7,11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,28)(25,29)(26,31)(30,35)(32,34)(33,36);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,28)(23,30)(25,33)(26,34)(27,35)(29,36)(31,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1995848);