#7855490

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

8419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2,3,4)(13,14,18,19)(15,16,17,20);(1,2)(3,4)(5,6,9,12)(7,8,11,10)(13,15,18,17)(14,16,19,20);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,19)(18,20);(13,15,18,17)(14,16,19,20);(5,7)(6,8)(9,11)(10,12)(13,17,18,15)(14,16,19,20);(5,8)(6,11)(7,12)(9,10)(13,16)(14,17)(15,19)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262314, #72262470, #72262476
maximal normal subgroup of	#72262314, #72262470, #72262476

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7762686);