#7712118

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

7110 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)(7,11)(8,10)(9,12)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(6,10)(9,11)(14,18)(15,21)(20,23)(22,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,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, 7619314);