#7919109

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

8340 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)(5,6,9,10)(7,8,11,12)(13,14,19,23)(15,22,24,18)(16,20,27,28)(17,25,26,21);(1,2)(3,4)(13,15)(14,18)(16,25)(17,20)(19,24)(21,27)(22,23)(26,28);(6,10)(7,11)(13,16,19,27)(14,20,23,28)(15,21,24,25)(17,18,26,22);(5,7)(6,8)(9,11)(10,12)(13,17,19,26)(14,21,23,25)(15,20,24,28)(16,18,27,22);(5,8)(6,7)(9,12)(10,11)(13,18)(14,15)(16,26)(17,27)(19,22)(20,21)(23,24)(25,28);(13,15)(14,22)(16,21)(17,20)(18,23)(19,24)(25,27)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261985, #72262101, #72262185
maximal normal subgroup of	#72261985, #72262101, #72262185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7826305);