#392614

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3458 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227325, #72227326, #72280919, …
maximal normal subgroup of	#72227326, #72280919, #72281084, …
maximal quotient of	#72281648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299810);