#7674174

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

6823 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581370);