#990956

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2350 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

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, 898152);