#6384016

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61038	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,f^d=fi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343474, #72343646, #72343648
maximal normal subgroup of	#72343474, #72343646, #72343648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291212);