#7674596

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

6994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=h,e^2=i,f^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, 7581792);