#1145568

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

2522 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fhi,e^2=fg,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, 1052764);