#7780042

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

6950 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)(13,14)(15,20)(16,22)(17,26)(18,23)(19,21)(24,28)(25,27);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,15,19,18)(14,20,21,23)(16,24,26,27)(17,25,22,28);(6,8)(10,12)(14,21)(15,18)(16,26)(25,28);(7,11)(10,12)(13,16,19,26)(14,22,21,17)(15,24,18,27)(20,28,23,25);(5,7)(6,10)(8,12)(9,11)(13,17,19,22)(14,16,21,26)(15,25,18,28)(20,24,23,27);(5,8,9,6)(7,12,11,10)(13,18,19,15)(14,23,21,20)(16,27,26,24)(17,28,22,25)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315169, #72315184, #72315203
maximal normal subgroup of	#72315169, #72315184, #72315203

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7687238);