#7674470

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

7214 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)(5,6,10,12)(7,11,9,8)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(1,2,3,4)(5,7,10,9)(6,11,12,8)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(6,12)(8,11)(14,15)(16,23)(20,22)(24,28);(5,8)(6,7)(9,12)(10,11)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(5,9,10,7)(6,8,12,11)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7581666);