#7739366

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8436 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312133, #72341476, #72341488
maximal normal subgroup of	#72312133, #72341476, #72341488

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646562);