#7631048

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

7506 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)(11,12)(14,19)(17,28)(20,26)(21,25);(5,6,10,7)(8,11,9,12)(13,14)(15,20)(16,25)(17,22)(18,19)(21,27)(23,26)(24,28);(1,2,3,4)(5,7,10,6)(8,12,9,11)(14,19)(15,23)(21,25)(22,24);(5,8,10,9)(6,11,7,12)(13,15,18,23)(14,20,19,26)(16,22,27,24)(17,21,28,25);(5,9,10,8)(6,12,7,11)(13,16,18,27)(14,21,19,25)(15,24,23,22)(17,26,28,20);(13,17)(14,22)(15,21)(16,26)(18,28)(19,24)(20,27)(23,25)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538244);