#7786190

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

7052 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)(7,11)(8,9)(10,12)(13,14,18,20)(15,19,21,28)(16,24,26,17)(22,23,25,27);(1,2,3,4)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27);(6,9)(10,11)(14,20)(15,21)(17,24)(22,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,17)(15,22)(18,26)(19,27)(20,24)(21,25)(23,28);(13,17)(14,16)(15,23)(18,24)(19,25)(20,26)(21,27)(22,28)

Polycyclic presentation:

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

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, 7693386);