#7883075

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

7069 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)(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,22)(17,27)(18,24)(19,21)(23,26)(25,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,26)(17,24)(18,27)(19,25)(21,28)(22,23);(5,6,9,8)(7,11,12,10)(14,21)(15,25)(16,24)(23,27);(13,16,19,24)(14,22,21,18)(15,26,25,17)(20,23,28,27);(5,7)(6,11)(8,10)(9,12)(13,17,19,26)(14,23,21,27)(15,24,25,16)(18,20,22,28);(5,8,9,6)(7,11,12,10)(13,18,19,22)(14,24,21,16)(15,27,25,23)(17,28,26,20)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,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	#72315015, #72315171, #72315191
maximal normal subgroup of	#72315015, #72315171, #72315191

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790271);