#7793510

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

7570 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)(13,14)(15,23)(16,22)(17,19)(18,20)(21,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,19,18)(14,20,17,23)(16,25,26,24)(21,28,22,27);(6,9)(10,11)(13,16,19,26)(14,21,17,22)(15,24,18,25)(20,27,23,28);(7,12)(10,11)(13,16)(14,22)(15,25)(17,21)(18,24)(19,26)(20,27)(23,28);(5,7,8,12)(6,10,9,11)(13,17,19,14)(15,20,18,23)(16,22,26,21)(24,27,25,28);(5,6)(7,10)(8,9)(11,12)(13,18,19,15)(14,20,17,23)(16,24,26,25)(21,28,22,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281540
maximal normal subgroup of	#72281540
maximal quotient of	#72282247

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7700706);