#356243

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
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

1384 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:

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

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^a=di,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224693, #72312233, #72312295, …
maximal normal subgroup of	#72224693, #72312233, #72312295, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263439);