#498318

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2213	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,e^b=ei,g^b=gi,f^c=fh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216402, #72216411, #72216414
maximal normal subgroup of	#72216402, #72216411, #72216414
maximal quotient of	#72226877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405514);