#2318405

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1774 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)(5,6,10,12)(7,9,11,8)(21,22)(23,28)(24,25)(26,27)(29,33)(30,35)(31,34)(32,36);(5,7,10,11)(6,8,12,9)(14,17)(18,20)(22,27)(23,30)(24,33)(32,34);(5,8,10,9)(6,11,12,7)(13,14,16,17)(15,18,19,20)(21,23,26,30)(22,28,27,35)(24,31,33,36)(25,34,29,32);(1,2,3,4)(5,9,10,8)(6,11,12,7)(21,24)(22,29)(23,31)(25,27)(26,33)(28,32)(30,36)(34,35);(5,6)(7,8)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25,26,29)(22,24,27,33)(23,32,30,34)(28,36,35,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=fi,e^2=gh,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, 2225601);