#2130529

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=fgh,e^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, 2037725);