#356080

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2346 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1757, #72342922, #72342941
maximal normal subgroup of	D₄×#1757, #72342922, #72342941
maximal quotient of	#72342986

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263276);