#7736396

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

8846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304554, #72304580, #72304592, …
maximal normal subgroup of	#72304554, #72304580, #72304592, …
maximal quotient of	#72336695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643592);