#7688200

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

6939 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315167, #72315179, #72315203
maximal normal subgroup of	#72315167, #72315179, #72315203

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7595396);