#392198

order	512
nilpotency class	3
isoclinism	isoclinic class of #32950
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

3360 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)(7,16)(8,20,23,29)(9,14)(10,15)(11,18)(12,22,28,21)(13,24)(17,31,34,25)(19,26)(27,35)(30,33,36,32);(5,7,9,18,10,19,24,35)(6,11,13,16,15,27,14,26)(8,17,21,32,23,34,22,33)(12,25,20,30,28,31,29,36);(5,8)(6,12)(7,17)(9,21)(10,23)(11,25)(13,20)(14,29)(15,28)(16,30)(18,32)(19,34)(22,24)(26,36)(27,31)(33,35);(2,4)(5,6)(7,11)(8,12)(9,13)(10,15)(14,24)(16,18)(17,25)(19,27)(20,21)(22,29)(23,28)(26,35)(30,32)(31,34)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227311, #72227326, #72253127, …
maximal normal subgroup of	#72227326, #72312172, #72312265, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299394);