#7672622

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

7351 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)(5,6,9,7)(8,10,12,11)(17,25)(20,24)(22,28)(26,27);(1,2)(3,4)(13,14,18,16)(15,19,23,21)(17,20,25,24)(22,26,28,27);(6,7)(10,11)(14,16)(19,21)(20,24)(26,27);(5,7,9,6)(8,11,12,10)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27);(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(5,8,9,12)(6,10,7,11)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579818);