#7630369

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

8078 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=g,f^2=i,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	#72280398, #72281628, #72304313, …
maximal normal subgroup of	#72280398, #72281628, #72304313, …
maximal quotient of	#72281105, #72282027, C₂×#7630369

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537565);