#7631826

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

8930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,f^2=h,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, 7539022);