#7631273

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

7612 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)(6,10)(7,12)(13,14)(15,23)(16,17)(18,27)(19,21)(20,24)(22,25)(26,28);(5,6,9,10)(7,11,12,8)(13,15)(14,20)(16,25)(17,27)(18,26)(19,24)(21,23)(22,28);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(14,21)(15,24)(16,26)(22,27);(5,8,9,11)(6,7,10,12)(13,16,19,26)(14,17,21,28)(15,25,24,18)(20,27,23,22);(13,17)(14,16)(15,22)(18,20)(19,28)(21,26)(23,25)(24,27);(13,18,19,25)(14,22,21,27)(15,26,24,16)(17,20,28,23)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,f^2=i,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, 7538469);