#7726015

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

8244 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)(13,14,18,21)(15,16,19,24)(17,25,26,20)(22,28,27,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,18,19)(14,16,21,24)(17,22,26,27)(20,23,25,28);(6,10)(7,11)(13,14)(15,16)(17,20)(18,21)(19,24)(22,23)(25,26)(27,28);(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,25)(26,28);(5,7,9,11)(6,8,10,12)(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,25)(24,28);(5,8)(6,11)(7,10)(9,12)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7633211);