#7985252

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

6837 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)(5,6)(7,8)(9,11)(10,12)(13,14,18,20)(15,21,16,19)(17,22,25,27)(23,28,24,26);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15,18,16)(14,19,20,21)(17,24,25,23)(22,28,27,26);(5,8,10,9)(6,7,12,11)(14,20)(15,16)(22,27)(23,24);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,26)(24,28)(25,27);(5,7)(6,8)(9,12)(10,11)(13,16,18,15)(14,21,20,19)(17,23,25,24)(22,26,27,28);(5,9,10,8)(6,11,12,7)(13,17,18,25)(14,22,20,27)(15,23,16,24)(19,26,21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315173, #72315192, #72315203
maximal normal subgroup of	#72315173, #72315192, #72315203

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892448);