#356242

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1394 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224693, #72227326, #72312252, …
maximal normal subgroup of	#72224693, #72227326, #72312252, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263438);