#7667101

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
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

7606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312837, #72315169, #72315195
maximal normal subgroup of	#72312837, #72315169, #72315195

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7574297);