#365174

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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1967 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)(5,6,9,14,11,8,10,13)(7,15,17,19,18,20,16,12);(1,3,4,2)(5,7,11,18)(6,12,8,19)(9,17,10,16)(13,20,14,15);(5,8)(6,11)(7,15)(9,14)(10,13)(12,17)(16,19)(18,20);(5,6)(7,12)(8,11)(9,14)(10,13)(15,17)(16,20)(18,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#24472, #72312219, #72312295
maximal normal subgroup of	C₄×#24472, #72312219, #72312295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272370);