#7986061

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

7573 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)(5,6,8,9)(7,10,12,11)(16,24)(20,27)(23,26)(25,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,22)(15,21,18,17)(16,20,24,27)(23,28,26,25);(6,9)(10,11)(13,15,19,18)(14,17,22,21)(16,23,24,26)(20,25,27,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,24)(21,28)(22,27);(5,6)(7,11)(8,9)(10,12)(13,17)(14,15)(16,25)(18,22)(19,21)(20,23)(24,28)(26,27);(5,7,8,12)(6,10,9,11)(13,18,19,15)(14,21,22,17)(16,26,24,23)(20,28,27,25)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,f^2=i,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	#72312319, #72312370, #72312421, …
maximal normal subgroup of	#72312319, #72312370, #72312421, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7893257);