#7630341

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

9232 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)(5,6,9,10)(7,11,8,12)(13,14,18,17)(15,16,20,19);(6,10)(11,12)(14,17)(15,20);(2,4)(13,15)(14,19)(16,17)(18,20);(5,7,9,8)(6,11,10,12)(13,16,18,19)(14,20,17,15);(5,7,9,8)(6,11,10,12)(13,17,18,14)(15,16,20,19);(5,8,9,7)(6,12,10,11)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304031, #41×#525, #72304532, …
maximal normal subgroup of	#72304031, #41×#525, #72304532, …
maximal quotient of	#72304051, #72305560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537537);