#371029

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1567 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)(7,10,15,11)(8,12,16,19)(9,13)(14,18)(17,20);(1,3,4,2)(5,7)(6,10,13,11)(8,14)(9,15)(12,17,19,18)(16,20);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19);(5,8)(6,12,13,19)(7,14)(9,16)(10,17,11,18)(15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#23996, #72245877, #72245896, …
maximal normal subgroup of	C₄×#23996, #72245877, #72245896, …
maximal quotient of	#72251461, #72269840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278225);