#185199

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92395);