#386835

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

834 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fg,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214501, #72216212, #72216747
maximal normal subgroup of	#72214501, #72216212, #72216747
maximal quotient of	#72214159

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294031);