#105536

order	512
nilpotency class	3
isoclinism	isoclinic class of #94966
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2)(3,4)(6,10)(7,8)(11,12)(13,14)(15,21,23,27)(16,19)(17,20)(18,22,25,28)(24,26);(5,6,8,11,9,12,7,10)(13,15)(14,18,20,25)(16,22)(17,23)(19,21,26,27)(24,28);(2,4)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240707, #72240740, #72240832, …
maximal normal subgroup of	#72240707, #72240740, #72240832, …
maximal quotient of	#72240713, #72240745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12732);