#7779696

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

7535 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)(13,14,18,19)(15,16,21,24)(17,20,25,26)(22,23,27,28);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28);(6,8)(10,12)(14,19)(15,21)(17,25)(23,28);(7,11)(10,12)(13,15,18,21)(14,16,19,24)(17,22,25,27)(20,23,26,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)(25,28)(26,27);(5,8,9,6)(7,12,11,10)(13,17,18,25)(14,20,19,26)(15,22,21,27)(16,23,24,28)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^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	#72297211
maximal normal subgroup of	#72297211
maximal quotient of	#72297258, #72297563, #72297978

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7686892);