#7630513

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

8616 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)(6,10)(11,12)(13,14)(15,19)(16,24)(17,21)(18,22)(20,25)(23,26)(27,28);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(15,22)(18,19)(23,28)(26,27);(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,27)(24,26)(25,28);(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,25)(26,28);(5,8,9,7)(6,12,10,11)(13,16,17,25)(14,20,21,24)(15,23,22,28)(18,26,19,27)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342299, #72342320, #72342329
maximal normal subgroup of	#72342299, #72342320, #72342329

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537709);