#7693694

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

7104 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)(5,6,10,8)(7,11,9,12)(13,14)(15,20)(16,25)(17,22)(18,21)(19,23)(24,28)(26,27);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15,19,17)(14,20,23,22)(16,26,18,24)(21,28,25,27);(6,8)(11,12)(13,16,19,18)(14,21,23,25)(15,24,17,26)(20,27,22,28);(5,8,10,6)(7,12,9,11)(13,17,19,15)(14,22,23,20)(16,24,18,26)(21,27,25,28);(5,9,10,7)(6,12,8,11)(13,14)(15,20)(16,21)(17,22)(18,25)(19,23)(24,27)(26,28);(13,18,19,16)(14,21,23,25)(15,24,17,26)(20,28,22,27)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,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	#72315015, #72315167, #72315185
maximal normal subgroup of	#72315015, #72315167, #72315185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7600890);