#209198

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

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×#449	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1107 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)(13,14,19,23)(15,24,16,21)(17,22,18,27)(20,26,28,25);(13,15,19,16)(14,20,23,28)(17,25,18,26)(21,27,24,22);(2,4)(13,16,19,15)(14,21,23,24)(17,25,18,26)(20,27,28,22);(5,7,8,12,9,6,10,11)(13,17,19,18)(14,22,23,27)(15,25,16,26)(20,21,28,24)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320625, #72320652, #72320669
maximal normal subgroup of	#72320625, #72320652, #72320669
maximal quotient of	#72321151, #72321376

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116394);