#368868

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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×#3068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1753 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)(6,12)(7,17)(8,20)(9,10)(13,18)(15,16);(1,3,4,2)(5,6)(7,18)(8,14)(9,16)(10,12)(11,15)(13,19)(17,20);(5,7,10,14,11,17,9,19)(6,13,12,18,15,8,16,20);(5,8)(6,14)(7,16)(9,18)(10,20)(11,13)(12,17)(15,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#24472, #72312223, #72312238
maximal normal subgroup of	C₄×#24472, #72312223, #72312238

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276064);