#7631042

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

8754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313631, #72342302, #72342346
maximal normal subgroup of	#72313631, #72342302, #72342346

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538238);