#719141

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6234 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280735, #72280790
maximal normal subgroup of	#72280735, #72280790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 626337);