#495725

order	512
nilpotency class	4
isoclinism	isoclinic class of #483873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33517	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2)(3,4)(5,6,9,13,10,15,19,14)(7,12,16,11,17,20,8,18);(6,11)(7,16)(8,17)(9,19)(12,13)(14,20)(15,18);(2,4);(5,7,9,16,10,17,19,8)(6,12,13,11,15,20,14,18)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213937, #72214893, #72216494, …
maximal normal subgroup of	#72213937, #72214893, #72216494, …
maximal quotient of	#72214781, #72216453, C₂×#495725, …
automorphism group of	#366, #367, #1368, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402921);