#2892838

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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

2012 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=hi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2800034);