#229171

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

740 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)(5,6,9,12,8,10,7,11)(14,18,21,25)(15,22)(20,27,28,23)(24,26);(5,6,7,11,8,10,9,12)(13,14,19,18,17,21,16,25)(15,23,26,28,22,27,24,20);(2,4)(6,10)(11,12);(5,7,8,9)(6,11,10,12)(13,15,19,26,17,22,16,24)(14,20,25,27,21,28,18,23)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343998, #72344001, #72344003
maximal normal subgroup of	#72343998, #72344001, #72344003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136367);