#7632474

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

8362 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539670);