#7630838

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

7826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|d^2=i,e^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, 7538034);