#7993367

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

7070 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)(13,14,18,19)(15,17,21,25)(16,20,23,26)(22,24,27,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(16,23)(20,26)(22,27)(24,28);(6,9)(10,12)(14,19)(15,21)(20,26)(22,27);(7,11)(10,12)(13,15)(14,17)(16,22)(18,21)(19,25)(20,24)(23,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,16,18,23)(14,20,19,26)(15,22,21,27)(17,24,25,28);(5,6)(7,10)(8,9)(11,12)(13,17)(14,15)(16,24)(18,25)(19,21)(20,22)(23,28)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7900563);