#6384252

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3318	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263948, #72263962, #72264037
maximal normal subgroup of	#72263948, #72263962, #72264037
maximal quotient of	#72265139

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291448);