#7942165

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

7848 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)(5,6,9,10)(7,8,11,12)(13,14)(15,22)(16,19)(17,24)(18,20)(21,25)(23,27)(26,28);(1,2)(3,4)(13,14,19,16)(15,22,18,20)(17,21,25,24)(23,28,26,27);(6,10)(7,11)(13,15,19,18)(14,20,16,22)(17,23,25,26)(21,27,24,28);(5,7)(6,8)(9,11)(10,12)(13,16,19,14)(15,20,18,22)(17,24,25,21)(23,27,26,28);(5,8)(6,7)(9,12)(10,11)(13,17)(14,21)(15,23)(16,24)(18,26)(19,25)(20,27)(22,28);(13,18,19,15)(14,20,16,22)(17,23,25,26)(21,28,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304329, C₂×#7942165
maximal normal subgroup of	#72304329, C₂×#7942165
maximal quotient of	C₂×#7942165

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7849361);