#7696657

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

7268 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)(5,6,9,10)(7,11,12,8)(13,14,19,18)(15,20,25,24)(16,26,27,21)(17,22,23,28);(1,2,3,4)(13,15)(14,20)(16,17)(18,24)(19,25)(21,28)(22,26)(23,27);(6,10)(7,12)(13,16,19,27)(14,21,18,26)(15,23,25,17)(20,22,24,28);(5,7)(6,11)(8,10)(9,12)(13,16)(14,21)(15,17)(18,26)(19,27)(20,28)(22,24)(23,25);(5,8,9,11)(6,7,10,12)(13,17)(14,22)(15,16)(18,28)(19,23)(20,26)(21,24)(25,27);(13,18,19,14)(15,24,25,20)(16,26,27,21)(17,22,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238190, #72238212, #72238214
maximal normal subgroup of	#72238190, #72238212, #72238214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603853);