#204202

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,e^b=eh,g^b=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309294, #72309297, #72309308
maximal normal subgroup of	#72309294, #72309297, #72309308

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111398);