#353127

order	512
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₄²
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁵	#41
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2398 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)(7,17,20,27)(8,13)(9,14)(10,16)(11,15)(12,19,26,21)(18,31,32,36)(22,28)(23,29)(24,30)(25,33,35,34);(2,4)(5,7)(6,12,15,26)(8,18)(9,19)(10,21)(11,20)(13,25,28,35)(14,17,16,27)(22,32)(23,33)(24,34)(29,31,30,36);(5,8,10,24,11,22,9,23)(6,13,16,30,15,28,14,29)(7,18,21,34,20,32,19,33)(12,25,27,36,26,35,17,31);(8,22)(13,28)(18,32)(23,24)(25,35)(29,30)(31,36)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	H(5,ℤ/4), #72210978, #72210987, …
maximal normal subgroup of	H(5,ℤ/4), #72210978, #72210987, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260323);