#7631854

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

8455 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)(6,7)(10,11)(13,14)(15,22)(16,19)(17,20)(18,21)(23,27)(24,25)(26,28);(5,6,9,7)(8,10,12,11)(13,15,18,19)(14,16,21,22)(17,25,26,23)(20,27,28,24);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,22)(16,19)(17,20)(18,21)(23,27)(24,25)(26,28);(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28);(13,16)(14,19)(15,21)(17,24)(18,22)(20,25)(23,28)(26,27);(5,8,9,12)(6,10,7,11)(13,17,18,26)(14,20,21,28)(15,23,19,25)(16,24,22,27)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539050);