#185067

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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×#3494	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2592 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)(3,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22)(23,27)(24,26)(25,29)(28,32)(30,34)(31,33)(35,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,23,25,30)(22,27,29,34)(24,31,32,36)(26,33,28,35);(23,30)(27,34)(31,36)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239068
maximal normal subgroup of	#72239068
maximal quotient of	#72239141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92263);