#376951

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,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	C₄×#24472, #72312221, #72312242
maximal normal subgroup of	C₄×#24472, #72312221, #72312242

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284147);