#7678274

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

7100 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)(10,12)(13,14)(15,17)(16,20)(18,22)(19,21)(23,26)(24,28)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,19,22,17)(16,24,27,23)(20,26,25,28);(2,4)(13,14,18,22)(15,17,21,19)(16,25,27,20)(23,28,24,26);(1,2)(3,4)(13,14)(15,19)(16,20)(17,21)(18,22)(23,28)(24,26)(25,27);(13,16,18,27)(14,20,22,25)(15,23,21,24)(17,26,19,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,22,15)(16,26,27,28)(20,24,25,23)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585470);