#7674127

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

8126 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)(6,9)(10,11)(13,14)(15,21)(16,18)(17,22)(19,20)(23,28)(24,25)(26,27);(5,6)(7,10)(8,9)(11,12)(14,20)(15,16)(22,27)(23,24);(2,4)(5,6)(7,11)(8,9)(10,12)(13,15,19,16)(14,21,20,18)(17,24,26,23)(22,25,27,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,16,19,15)(14,18,20,21)(17,24,26,23)(22,25,27,28);(13,17,19,26)(14,22,20,27)(15,23,16,24)(18,25,21,28);(13,18)(14,15)(16,20)(17,25)(19,21)(22,23)(24,27)(26,28)

Polycyclic presentation:

2^9|c^2=i,d^2=g,e^2=i,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, 7581323);