#182631

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,e^2=h,b^a=be,c^a=ch,e^a=eh,g^a=ghi,d^b=dh,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257787, #72264049
maximal normal subgroup of	#72257787, #72264049
maximal quotient of	#72264458

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89827);