#7631868

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

7976 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)(6,10)(7,12)(13,14,18,17)(15,21,24,19)(16,20,26,25)(22,28,23,27);(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,23)(17,21)(18,24)(20,28)(22,26)(25,27);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,16)(14,20)(15,22)(17,25)(18,26)(19,27)(21,28)(23,24);(13,16)(14,20)(15,23)(17,25)(18,26)(19,28)(21,27)(22,24);(13,15)(14,21)(16,22)(17,19)(18,24)(20,28)(23,26)(25,27);(5,8,9,11)(6,12,10,7)(13,17,18,14)(15,19,24,21)(16,25,26,20)(22,27,23,28)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,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, 7539064);