#388074

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efg,d^2=efg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72235146, #72235149, #72235153

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295270);