#7657193

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564389);