#790461

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 697657);