#7631845

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

8386 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)(5,6,9,12)(7,10,8,11)(13,14,19,16)(15,23,25,20)(17,21,24,26)(18,22,28,27);(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,23)(17,18)(19,25)(21,27)(22,26)(24,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,16)(15,23,25,20)(17,21,24,26)(18,22,28,27);(5,7,9,8)(6,10,12,11)(13,16,19,14)(15,23,25,20)(17,21,24,26)(18,27,28,22);(5,8,9,7)(6,11,12,10)(13,17,19,24)(14,21,16,26)(15,18,25,28)(20,27,23,22);(13,18)(14,22)(15,24)(16,27)(17,25)(19,28)(20,21)(23,26)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=g,e^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, 7539041);