#7718213

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

7638 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)(8,12)(10,11)(13,14)(15,21)(16,25)(17,27)(18,20)(19,22)(23,28)(24,26);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,26)(17,24)(18,22)(20,21)(23,27)(25,28);(6,7)(10,11)(14,20)(15,22)(16,27)(24,28);(13,15)(14,21)(16,23)(17,24)(18,22)(19,20)(25,28)(26,27);(5,7,9,6)(8,11,12,10)(13,16)(14,17)(15,23)(18,27)(19,28)(20,25)(21,24)(22,26);(5,8)(6,10)(7,11)(9,12)(13,17)(14,16)(15,24)(18,25)(19,26)(20,27)(21,23)(22,28)

Polycyclic presentation:

2^9|b^2=gh,e^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7625409);