#7711923

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

7344 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)(7,11)(10,12)(14,19)(16,26)(17,24)(20,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,20,25,23)(16,21,26,27)(17,22,24,28);(6,9)(10,12)(13,14)(15,23)(16,27)(17,22)(18,19)(20,25)(21,26)(24,28);(13,15,18,25)(14,20,19,23)(16,24,26,17)(21,28,27,22);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,24)(17,25)(18,26)(19,27)(20,28)(22,23);(5,7,8,11)(6,10,9,12)(13,17,18,24)(14,22,19,28)(15,16,25,26)(20,21,23,27)

Polycyclic presentation:

2^9|b^2=h,d^2=h,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, 7619119);