#7631423

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

8798 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)(14,17)(18,20);(1,2)(3,4)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20);(7,11)(10,12)(13,15,16,19)(14,18,17,20);(5,7)(6,10)(8,11)(9,12);(5,6)(7,10)(8,9)(11,12)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233756, #72233767, #72234456
maximal normal subgroup of	#72233756, #72233767, #72234456

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538619);