#146629

order	512
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×Q₈	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250630, #72261939, #72261982
maximal normal subgroup of	#72250630, #72261939, #72261982
maximal quotient of	#72250634, #72250636

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53825);