#497772

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319320, #72319574, #72319645
maximal normal subgroup of	#72319320, #72319574, #72319645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404968);