#229111

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 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)(5,6,8,11,9,7,10,12)(14,19,21,26)(16,23)(20,27,28,24)(22,25);(5,7,8,12,9,6,10,11)(13,14,15,19,18,21,17,26)(16,24,22,28,23,27,25,20);(2,4)(13,15,18,17)(14,19,21,26)(16,22,23,25)(20,24,28,27);(13,16,17,25,18,23,15,22)(14,20,19,24,21,28,26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318777, #72318785, #72318791, …
maximal normal subgroup of	#72318777, #72318785, #72318791, …
maximal quotient of	#72322520

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136307);