#108811

order	512
nilpotency class	3
isoclinism	isoclinic class of #94972
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1100 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240707, #72240803, #72257797, …
maximal normal subgroup of	#72240707, #72240803, #72257797, …
maximal quotient of	#72240712, #72240721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16007);