#513175

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2263, #72342897
maximal normal subgroup of	D₄×#2263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420371);