#7632464

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

8225 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)(15,19)(16,25)(17,22)(18,21)(20,23)(24,28)(26,27);(5,6,9,10)(7,11,12,8)(13,15,20,18)(14,21,23,19)(16,26,17,24)(22,27,25,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,20,17)(14,22,23,25)(15,24,18,26)(19,27,21,28);(13,17,20,16)(14,22,23,25)(15,24,18,26)(19,28,21,27);(13,18,20,15)(14,21,23,19)(16,26,17,24)(22,28,25,27);(5,8,9,11)(6,7,10,12)(13,19,20,21)(14,15,23,18)(16,27,17,28)(22,24,25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539660);