#195139

order	512
nilpotency class	3
isoclinism	isoclinic class of #18132
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1364 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240266, #72240337, #72240340
maximal normal subgroup of	#72240266, #72240337, #72240340

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102335);