#7626000

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,26)(24,30)(25,29)(27,32)(28,33)(31,35)(34,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,27)(26,32)(28,34)(33,36);(6,12)(10,13)(11,20)(18,19)(23,29)(26,32)(30,35)(33,36);(22,27)(26,32)(28,34)(33,36);(21,24,25,31)(22,28,27,34)(23,30,29,35)(26,33,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533196);