#7870505

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

7485 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)(7,8)(9,10)(11,12)(13,14)(15,20)(16,19)(17,21)(18,22)(23,27)(24,26)(25,28);(1,2)(3,4)(5,7,11,9)(6,8,12,10)(13,15,18,16)(14,19,22,20)(17,23,25,24)(21,26,28,27);(5,8,11,10)(6,9,12,7)(13,14,18,22)(15,19,16,20)(17,21,25,28)(23,26,24,27);(5,7,11,9)(6,8,12,10)(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28);(5,9,11,7)(6,8,12,10)(13,16,18,15)(14,20,22,19)(17,24,25,23)(21,27,28,26);(5,10,11,8)(6,9,12,7)(13,17,18,25)(14,21,22,28)(15,23,16,24)(19,26,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312837, #72312916, #72315199
maximal normal subgroup of	#72312837, #72312916, #72315199

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7777701);