#93666

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₈×C₁₆ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₈×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂²
socle, prime radical	C₂²	#1311
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 48 points:

(17,18,19,24,20,26,29,39,21,27,30,37,31,41,43,47,22,25,28,38,32,40,42,48,33,23,34,35,44,36,45,46);(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,25,27)(19,28)(24,37,38,35)(26,36,40,41)(29,42)(30,34)(39,47,48,46)(43,45)

Polycyclic presentation:

2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bg,d^a=di,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72230692, #72230696, #72230718, …
maximal subgroup of	#72216226, #72225140, #72225369, …
maximal normal subgroup of	#72216226, #72225140, #72225369, …
maximal quotient of	#72225071, #72230688, C₂×#93666, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 862);