#7670728

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

8238 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234905, #72234906, #72237369
maximal normal subgroup of	#72234905, #72234906, #72237369

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577924);