#1125668

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

2323 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1032864);