#7692334

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

7291 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)(5,6,9,12)(7,10,8,11)(13,14)(15,24)(16,17)(18,22)(19,21)(20,23)(25,26)(27,28);(5,6)(7,10)(8,11)(9,12)(13,15,20,19)(14,21,23,24)(16,27,18,25)(17,26,22,28);(1,2,3,4)(13,16,20,18)(14,22,23,17)(15,25,19,27)(21,26,24,28);(1,2)(3,4)(13,17,20,22)(14,16,23,18)(15,26,19,28)(21,27,24,25);(5,7,9,8)(6,10,12,11)(13,18,20,16)(14,22,23,17)(15,25,19,27)(21,28,24,26);(5,8,9,7)(6,11,12,10)(13,19,20,15)(14,21,23,24)(16,27,18,25)(17,28,22,26)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=i,e^2=g,f^2=g,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	#72238172, #72238212, #72238245
maximal normal subgroup of	#72238172, #72238212, #72238245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7599530);