#7630815

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

8102 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)(6,9)(10,12)(14,20)(18,22)(21,24)(23,28);(5,6)(7,10)(8,9)(11,12)(13,14,19,20)(15,24,26,21)(16,22,25,18)(17,23,27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,21,20,24)(16,17,25,27)(18,23,22,28);(13,16)(14,22)(15,17)(18,20)(19,25)(21,28)(23,24)(26,27);(13,17)(14,23)(15,25)(16,26)(18,24)(19,27)(20,28)(21,22);(5,7,8,11)(6,10,9,12)(13,18,19,22)(14,16,20,25)(15,23,26,28)(17,24,27,21)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538011);