#7966644

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

8505 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)(5,6)(7,11)(8,9)(10,12)(13,14,17,18)(15,21,22,19)(16,20,26,24)(23,28,25,27);(1,2)(3,4)(6,9)(10,11)(13,14)(15,19)(16,24)(17,18)(20,26)(21,22)(23,28)(25,27);(5,6)(7,10)(8,9)(11,12)(14,18)(15,22)(20,24)(23,25);(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28);(5,7)(6,11)(8,12)(9,10)(13,15,17,22)(14,19,18,21)(16,25,26,23)(20,28,24,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,23)(17,26)(18,24)(19,27)(21,28)(22,25)

Polycyclic presentation:

2^9|a^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304527, #72304537, #72304549
maximal normal subgroup of	#72304527, #72304537, #72304549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7873840);