#6382159

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34935	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2400 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)(3,4)(6,10)(11,12)(13,14,18,20,17,21,16,25)(15,19,24,26,23,27,22,28);(5,6,9,10)(7,11,8,12)(13,14,16,25,17,21,18,20)(15,19,22,28,23,27,24,26);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,23)(14,19,21,27)(16,22,18,24)(20,26,25,28);(5,8,9,7)(6,11,10,12)(13,16,17,18)(14,20,21,25)(15,22,23,24)(19,26,27,28);(14,21)(15,23)(20,25)(22,24)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,d^b=df,e^b=ei,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307418, #72307613
maximal normal subgroup of	#72307418, #72307613

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289355);