#7631865

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

7938 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);(5,7,9,11)(6,10,12,8)(13,14)(15,22)(16,19)(17,25)(18,21)(20,26)(23,24)(27,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,18,19)(14,16,21,22)(17,23,26,27)(20,24,25,28);(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,26)(27,28);(13,16)(14,19)(15,21)(17,24)(18,22)(20,27)(23,25)(26,28);(5,8,9,10)(6,11,12,7)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^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, 7539061);