#7732408

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

7432 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238159, #72238163, #72238255
maximal normal subgroup of	#72238159, #72238163, #72238255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7639604);