#6389184

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
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×#3062	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2176 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,22)(16,20)(17,27)(18,21)(19,25)(23,24)(26,28);(1,3,4,2)(5,7)(6,9)(8,12)(10,11);(13,15,17,25,18,26,16,24)(14,19,20,22,21,23,27,28);(7,12)(9,11)(13,14)(15,23)(16,27)(17,20)(18,21)(19,26)(22,24)(25,28);(6,10)(9,11)(13,16,18,17)(14,20,21,27)(15,24,26,25)(19,22,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240753, #72257163, #72257224
maximal normal subgroup of	#72240753, #72257163, #72257224
maximal quotient of	#72240778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296380);