#7630585

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

7848 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)(6,9)(10,12)(13,14)(15,24)(16,22)(17,21)(18,20)(19,23)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,26)(18,25)(19,21)(20,27)(22,28)(23,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,18)(14,20,23,22)(15,25,21,26)(17,28,24,27);(13,17,19,24)(14,21,23,15)(16,27,18,28)(20,25,22,26);(13,14)(15,24)(16,20)(17,21)(18,22)(19,23)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,18,19,16)(14,22,23,20)(15,25,21,26)(17,28,24,27)

Polycyclic presentation:

2^9|c^2=i,d^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537781);