#185098

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239885
maximal normal subgroup of	#72239885
maximal quotient of	#72239899

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92294);