#7631099

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

8430 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)(6,10)(11,12)(13,14)(15,20)(16,18)(17,22)(19,21)(23,27)(24,25)(26,28);(1,2)(3,4)(13,15,19,16)(14,20,21,18)(17,24,26,23)(22,25,28,27);(5,6)(7,11)(8,12)(9,10)(14,21)(15,16)(22,28)(23,24);(5,7,9,8)(6,11,10,12)(13,16,19,15)(14,18,21,20)(17,23,26,24)(22,27,28,25);(5,8,9,7)(6,12,10,11)(13,17,19,26)(14,22,21,28)(15,23,16,24)(18,25,20,27);(13,18)(14,15)(16,21)(17,25)(19,20)(22,23)(24,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538295);