#7987484

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

7180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^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	#72312575, #72315207, #72315209
maximal normal subgroup of	#72312575, #72315207, #72315209

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7894680);