#7631123

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

8802 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)(16,24)(19,26)(22,28)(25,27);(1,2)(3,4)(5,6,10,7)(8,11,9,12)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(5,7,10,6)(8,12,9,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(5,8,10,9)(6,11,7,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,9,10,8)(6,12,7,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,e^2=h,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	#72342302, #72342339, #72342341
maximal normal subgroup of	#72342302, #72342339, #72342341

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538319);