#225633

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fh,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309513, #72309517, #72309567, …
maximal normal subgroup of	#72309513, #72309517, #72309567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132829);