#5025757

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 2097152
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

2052 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)(6,8)(7,11)(14,17)(15,19)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34);(5,6,9,8)(7,10,11,12)(23,29)(26,33)(30,35)(34,36);(6,8)(10,12)(13,14,16,17)(15,20,19,18)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15,16,19)(14,18,17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(5,8,9,6)(7,10,11,12)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=g,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=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, 4932953);